The determination of gas and liquid flow rates in gas-liquid mixtures are important measurements in the oil and gas industry.
An example of an apparatus for measuring such flow rates is Schlumberger's PhaseTester™ VenturiX™ system (see e.g. I. Atkinson, M. Berard, B-V Hanssen, G. Ségéral, 17th International North Sea Flow Measurement Workshop, Oslo, Norway 25-28 Oct. 1999 “New Generation Multiphase Flowmeters from Schlumberger and Framo Engineering AS”; and http://www.slb.com/oilfield/index.cfm?id=id32270) which comprises a vertically mounted Venturi flow meter, a dual energy gamma-ray hold up measuring device and associated processors. This system successfully allows the simultaneous calculation of gas, water and oil volumetric flow rates in multi phase flows.
However, with conventional implementations of VenturiX™ technology the accuracy of the calculations starts to degrade as the gas volume fraction (GVF) increases above about 85% (the GVF being defined as the gas volumetric flow rate divided by the total volumetric flow rate of the gas-liquid mixture). This can be a problem because as oil wells age the GVF increases towards 100% and as gas wells age the GVF decreases from 100%. One reason for the drop in accuracy is that at low mixture densities (i.e. high GVFs), the accuracy of high energy gamma-ray density measurements starts to fall. A more fundamental reason, however, is that an underlying assumption of the calculations starts to break down at high GVFs. This assumption is that the total mass flow rate of the mixture is approximately equal to the liquid mass flow rate. The problem becomes particularly severe above about a GVF of 95%. Under flow conditions typical in North Sea natural gas production, a GVF of 95% approximately corresponds to the liquid mass flow rate being equal to the gas mass flow rate. Shell Expro have defined the wet gas range as a flow with a GVF>95% (R. N. Steven, “Wet Gas Metering with a Horizontally Mounted Venturi Meter”, Flow Measurement and Instrumentation, 2002, 361-372).
One approach to overcome this problem is to separate the fluid mixture into gas and liquid phases and measure the flow rate of each phase using conventional flow metering techniques. However, this relies on the efficient separation of the gas and liquid phases, which is difficult to perform reliably. Furthermore the equipment tends to be bulky and expensive.
Other investigators (e.g. Steven ibid.; and Z. H. Lin, “Two-Phase Flow Measurements with Orifices”, Encyclopaedia of Fluid Mechanics, Chapter 29, Vol. 3, Gulf, 1986) have proposed expressions for calculating the flow rate of a multi phase mixture through an orifice plate or a Venturi flow meter. The aim of most has been to find a universal expression/experimental correlation for calculating the flow rate at all GVF values. Although many expressions have been proposed, there is no agreement as to which is the most accurate. However, the differences between the correlations are small when they are used to calculate the flow rate of a wet gas. Steven ibid. provides a summary of two wet gas correlations for horizontal Venturi flow meters and five for orifice plate flow meters. The correlations assume that the flows are incompressible, there are no appreciable thermodynamic effects and the liquid flow rate is initially known.
The correlations are all based on the principle of relating the gas mass flow rate, Qg, to a “pseudo single phase gas mass flow rate”, Qtp, calculated from the standard Venturi/orifice plate equation using the measured differential pressure, ΔP, and the gas density, ρg:Qtp=KgAT√{square root over (2ρgΔP)}Qg=f(Qtp, Ql/Qg)where AT is the Venturi throat cross-sectional area, Kg is a function of the discharge coefficient and Venturi dimensions, and Ql is the liquid mass flow rate.
Alternatively, the correlations can, of course, be expressed in terms of a “pseudo single phase gas volume flow rate”, qtp:
            q      tp        =                  K        g            ⁢              A        T            ⁢                                    2            ⁢                                                  ⁢            Δ            ⁢                                                  ⁢                          P              tp                                            ρ            g                                          q      g        =          f      ⁡              (                              q            tp                    ,                                    q              l                        /                          q              g                                      )            
Essentially, correcting Qtp or qtp for multi phase flow based on the relative gas/liquid phase content gives the gas flow rate. However, in order to perform this correction the correlations require an additional input. A number of investigators have used the liquid flow rate, which can be measured using a tracer dilution technique (see e.g. N. Nederveen, G. V. Washington, F. H. Batstra, “Wet Gas Flow Measurement” SPE 19077, 1989; R. de Leeuw, “Liquid Correction of Venturi Meter Readings in Wet Gas Flow”, North Sea Flow Measurement Workshop, Norway, 1997; A. B. Al-Taweel, S. G. Barlow, “Wellsite Wet Gas Measurement System in Saudi Arabia” SPE 49162, 1998; and M. R. Konopczynski, H. de Leeuw, “Large Scale Application of Wet Gas Metering at the Oman Upstream LNG Project”, SPE 62119, 2000). However, tracer measurements involve practical difficulties and can be inconvenient to perform. For example, an upstream location has to be adapted and made available for tracer injection and usually a technician has to be present. Moreover tracer measurements are typically non-continuous.
Another approach is embodied in the Solartron ISA Dualstream II™ differential pressure based metering system. The system comprises three stages: a proprietary upstream flow conditioner, a classical Venturi flow meter and a second proprietary differential pressure flow meter. The system operates on the principle that the second differential pressure flow meter exhibits a significantly different response to the presence of liquid in the gas stream than the Venturi flow meter. Consequently the two flow meters provide two independent simultaneous equations derived from a wet gas correlation that can be solved to give the gas and liquid flow rates.
Notation
The following notation is used herein:    q=volumetric flow rate (m3/s)    Q=mass flow rate (kg/s)    ρ=density (kg/m3)    η=dynamic viscosity (Pa·s)    ε=gas expansivity    ΔP=differential pressure across the Venturi (Pa)    GVF=gas volume fraction (i.e. gas volumetric flow rate divided by the total volumetric flow rate)    α=hold up    wlr=water liquid ratio=qw/ql     M=Murdock coefficient    C=discharge coefficient    AT=cross-sectional area of Venturi throat (m2)    d=Venturi throat diameter (m)    D=Venturi inlet diameter (m)    h=distance between differential pressure tappings (m)    β=d/D    E=1/(1−β4)0.5     K=Flow coefficient (Kl=ClE and Kg=CgEε)    g=acceleration due to gravity (m2/s)    V=superficial velocity (m/s)    Re=Reynolds Number    Fr=Froude Number
  X  =            Lockhart      ⁢              -            ⁢      Martinelli      ⁢                          ⁢      parameter        =                                        q            l                                q            g                          ⁢                                            ρ              l                                      ρ              g                                          =                                    Q            l                                Q            g                          ⁢                                            ρ              g                                      ρ              l                                          
Subscripts:    l=liquid    g=gas    o=oil    w=water    m=mixture    tp=pseudo single phase